Embedability between right-angled Artin groups

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph \gam, we produce a new graph through a purely combinatorial procedure, and call it the extension graph \gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of \gam^e, by adding extra vertices for each complete subgraph of \gam^e. We prove that each finite induced subgraph $\Lambda$ of \gam^e gives rise to an inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an inclusion A(\Lambda)\to A(\gam) then $\Lambda$ is an induced subgraph of \gam^e_k. These results have a number of corollaries. Let $P_4$ denote the path on four vertices and let $C_n$ denote the cycle of length $n$. We prove that $A(P_4)$ embeds in A(\gam) if and only if $P_4$ is an induced subgraph of \gam. We prove that if $F$ is any finite forest then $A(F)$ embeds in $A(P_4)$. We recover the first author's result on co--contraction of graphs and prove that if \gam has no triangles and A(\gam) contains a copy of $A(C_n)$ for some $n\geq 5$, then \gam contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites' Theorem, which asserts that if $A(C_4)$ embeds in A(\gam) then \gam contains an induced square. Finally, we determine precisely when there is an inclusion $A(C_m)\to A(C_n)$ and show that there is no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is quasi-isometric to a tre

Geodesic acoustic modes in a fluid model of tokamak plasma : the effects of finite beta and collisionality

Starting from the Braginskii equations, relevant for the tokamak edge region, a complete set of nonlinear equations for the geodesic acoustic modes (GAM) has been derived which includes collisionality, plasma beta and external sources of particle, momentum and heat. Local linear analysis shows that the GAM frequency increases with collisionality at low radial wave number $k_{r}$ and decreases at high $k_{r}$. GAM frequency also decreases with plasma beta. Radial profiles of GAM frequency for two Tore Supra shots, which were part of a collisionality scan, are compared with these calculations. Discrepency between experiment and theory is observed, which seems to be explained by a finite $k_{r}$ for the GAM when flux surface averaged density $\langle n \rangle$ and temperature $\langle T \rangle$ are assumed to vanish. It is shown that this agreement is incidental and self-consistent inclusion of $\langle n \rangle$ and $\langle T \rangle$ responses enhances the disagreement more with $k_r$ at high $k_{r}$ . So the discrepancy between the linear GAM calculation, (which persist also for more "complete" linear models such as gyrokinetics) can probably not be resolved by simply adding a finite $k_{r}$

Effects of the Second Harmonic and Plasma Shaping on the Geodesic Acoustic Mode

The effects of second harmonics of the density and temperature perturbations on the linear Geodesic Acoustic Mode (GAM) frequency and non-linear generation of the GAM are investigated, using a fluid model. We show that the second harmonics contribute to the frequency through the density gradient scale length and the wave number of the GAM. In addition, the linear frequency of the GAM is generally increased by coupling to the higher harmonic.Comment: 4 pages, 3 figures, 41st EPS Conference Berlin 201

Structural basis for the inhibition of RecBCD by Gam and its synergistic antibacterial effect with quinolones

Our previous paper (Wilkinson et al, 2016) used high-resolution cryo-electron microscopy to solve the structure of the Escherichia coli RecBCD complex, which acts in both the repair of double-stranded DNA breaks and the degradation of bacteriophage DNA. To counteract the latter activity, bacteriophage λ encodes a small protein inhibitor called Gam that binds to RecBCD and inactivates the complex. Here, we show that Gam inhibits RecBCD by competing at the DNA-binding site. The interaction surface is extensive and involves molecular mimicry of the DNA substrate. We also show that expression of Gam in E. coli or Klebsiella pneumoniae increases sensitivity to fluoroquinolones; antibacterials that kill cells by inhibiting topoisomerases and inducing double-stranded DNA breaks. Furthermore, fluoroquinolone-resistance in K. pneumoniae clinical isolates is reversed by expression of Gam. Together, our data explain the synthetic lethality observed between topoisomerase-induced DNA breaks and the RecBCD gene products, suggesting a new co-antibacterial strategy

Strongly Coupled Matter-Field and Non-Analytic Decay Rate of Dipole Molecules in a Waveguide

The decay rate \gam of an excited dipole molecule inside a waveguide is evaluated for the strongly coupled matter-field case near a cutoff frequency \ome_c without using perturbation analysis. Due to the singularity in the density of photon states at the cutoff frequency, we find that \gam depends non-analytically on the coupling constant $\ggg$ as $\ggg^{4/3}$. In contrast to the ordinary evaluation of \gam which relies on the Fermi golden rule (itself based on perturbation analysis), \gam has an upper bound and does not diverge at \ome_c even if we assume perfect conductance in the waveguide walls. As a result, again in contrast to the statement found in the literature, the speed of emitted light from the molecule does not vanish at \ome_c and is proportional to $c\ggg^{2/3}$ which is on the order of $10^3 \sim 10^4$ m/s for typical dipole molecules.Comment: 4 pages, 2 figure